Goal: Find the equation, vertices, and foci of an ellipse. 8.2 Ellipses

What you’ll learn about Geometry of an Ellipse … and why Ellipses are the paths of planets and comets around the Sun, or of moons around planets.

Elliptical Orbits Around the Sun

Hale-Bopp Comet (1997)

prolate

Washington D.C. (pg. 598)

Paris Subway

Lithotripter (pg. 598)

Key Points on the Focal Axis of an Ellipse Note: The foci and vertices always lie on the major axis.

center: critical points: endpoints of major and minor axes length of horizontal axis: length of vertical axis: focal radius (distance from center to a focus):

Example: Fill in the blanks and graph. 1. center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

Example: Fill in the blanks and graph. 2. 𝑥 2 49 + 𝑦 2 36 =1 center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

Example: Fill in the blanks and graph. 3. 𝑥+1 2 4 + 𝑦−1 2 9 =1 center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

Example: Finding an Equation of an Ellipse The center is (0, 0). The length of the major axis (on the x-axis) is 6, and the length of the minor axis is 2. Equation:

Example: Finding an Equation of an Ellipse The center is (-2, 3). The length of the major axis (parallel to the x-axis) is 14, and the length of the minor axis is 6. Equation:

Example: Finding an Equation of an Ellipse

Example: Find the Vertices and Foci of the Ellipse

Example: Find the Equation in Standard Form and Graph x2 + 2x + 9y2 – 54y + 73 = 0